{"problem":{"name":"I Hate Non-integer Number","description":{"content":"You are given a sequence of positive integers $A=(a_1,\\ldots,a_N)$ of length $N$.   There are $(2^N-1)$ ways to choose one or more terms of $A$. How many of them have an integer-valued average? Find t","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2500,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"abc262_d"},"statements":[{"statement_type":"Markdown","content":"You are given a sequence of positive integers $A=(a_1,\\ldots,a_N)$ of length $N$.  \nThere are $(2^N-1)$ ways to choose one or more terms of $A$. How many of them have an integer-valued average? Find the count modulo $998244353$.\n\n## Constraints\n\n*   $1 \\leq N \\leq 100$\n*   $1 \\leq a_i \\leq 10^9$\n*   All values in input are integers.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$\n$a_1$ $\\ldots$ $a_N$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"abc262_d","tags":[],"sample_group":[["3\n2 6 2","6\n\nFor each way to choose terms of $A$, the average is obtained as follows:\n\n*   If just $a_1$ is chosen, the average is $\\frac{a_1}{1}=\\frac{2}{1} = 2$, which is an integer.\n    \n*   If just $a_2$ is chosen, the average is $\\frac{a_2}{1}=\\frac{6}{1} = 6$, which is an integer.\n    \n*   If just $a_3$ is chosen, the average is $\\frac{a_3}{1}=\\frac{2}{1} = 2$, which is an integer.\n    \n*   If $a_1$ and $a_2$ are chosen, the average is $\\frac{a_1+a_2}{2}=\\frac{2+6}{2} = 4$, which is an integer.\n    \n*   If $a_1$ and $a_3$ are chosen, the average is $\\frac{a_1+a_3}{2}=\\frac{2+2}{2} = 2$, which is an integer.\n    \n*   If $a_2$ and $a_3$ are chosen, the average is $\\frac{a_2+a_3}{2}=\\frac{6+2}{2} = 4$, which is an integer.\n    \n*   If $a_1$, $a_2$, and $a_3$ are chosen, the average is $\\frac{a_1+a_2+a_3}{3}=\\frac{2+6+2}{3} = \\frac{10}{3}$, which is not an integer.\n    \n\nTherefore, $6$ ways satisfy the condition."],["5\n5 5 5 5 5","31\n\nRegardless of the choice of one or more terms of $A$, the average equals $5$."]],"created_at":"2026-03-03 11:01:14"}}